Elsevier

Journal of Theoretical Biology

Volume 425, 21 July 2017, Pages 53-71
Journal of Theoretical Biology

Mathematical models of retinitis pigmentosa: The oxygen toxicity hypothesis

https://doi.org/10.1016/j.jtbi.2017.05.006Get rights and content

Highlights

  • Hyperoxia can explain some of the patterns of photoreceptor degeneration seen in vivo.

  • Patches of retinal loss will grow if photoreceptor density is low at their boundaries.

  • The wave speed of retinal degeneration decreases as photoreceptor density increases.

  • Treatment with antioxidants and trophic factors may limit photoreceptor degeneration.

  • Capillary loss may limit photoreceptor degeneration.

Abstract

The group of genetically mediated diseases, known collectively as retinitis pigmentosa (RP), cause retinal degeneration and, hence, loss of vision. The most common inherited retinal degeneration, RP is currently untreatable. The retina detects light using cells known as photoreceptors, of which there are two types: rods and cones. In RP, genetic mutations cause patches of photoreceptors to degenerate and typically directly affect either rods or cones, but not both. During disease progression, degenerate patches spread and the unaffected photoreceptor type also begins to degenerate. The cause underlying these phenomena is currently unknown. The oxygen toxicity hypothesis proposes that secondary photoreceptor loss is due to hyperoxia (toxically high oxygen levels), which results from the decrease in oxygen uptake following the initial loss of photoreceptors. In this paper, we construct mathematical models, formulated as 1D systems of partial differential equations, to investigate this hypothesis. Using a combination of numerical simulations, asymptotic analysis and travelling wave analysis, we find that degeneration may spread due to hyperoxia, and generate spatio-temporal patterns of degeneration similar to those seen in vivo. We determine the conditions under which a degenerate patch will spread and show that the wave speed of degeneration is a monotone decreasing function of the local photoreceptor density. Lastly, the effects of treatment with antioxidants and trophic factors, and of capillary loss, upon the dynamics of photoreceptor loss and recovery are considered.

Introduction

The term retinitis pigmentosa (RP) refers to a group of genetically mediated retinal degenerative diseases that cause a progressive loss of visual function. RP is the most common inherited retinal degeneration, with a prevalence (in its nonsyndromic form) of 1 in 4000, corresponding to a total of 1.5 million affected individuals worldwide (Hamel, 2006, Hartong, Berson, Dryja, 2006, Shintani, Shechtman, Gurwood, 2009). At present no treatments are clinically available either to slow its progression or reverse its effects (Musarella and MacDonald, 2011). In this paper, we investigate the prospective role of oxygen toxicity (or hyperoxia) in the progression of RP.

The retina is the innermost (closest to the centre of the eye) tissue layer in the eye, extending from the optic disc, where the optic nerve, central retinal artery and vein puncture the retina, to the ora serrata (Fig. 1(a)). The outer (furthest from the centre of the eye) layer of the retina, with which we are concerned here, is mainly populated by two types of cell: photoreceptors and retinal pigment epithelium (RPE) cells. The light-detecting photoreceptors come in two varieties: rods, which confer achromatic vision under scotopic (low light) conditions and cones, which provide high-acuity colour vision under photopic (well-lit) conditions.

The outer parts of the photoreceptors are composed of inner segments (ISs) and outer segments (OSs). The ISs contain most of a photoreceptor’s mitochondria, constituting their primary site of oxygen consumption, whilst the OSs are composed of discs in which light-sensitive photopigments are embedded. Discs are periodically shed from the tips of OSs, where they are phagocytosed by the overlying RPE cells, and are regenerated at the OS base, with a turnover rate of 9–13 days (Young, 1971, see also Oyster, 1999, Young, 1967, Young, 1978, Young, Bok, 1969.

The average human retina contains 92 million rods and 4.6 million cones (Curcio et al., 1990). Fig. 1(b) shows the distribution of rods and cones in a typical human retina. Cones attain their highest density in a sharp peak centred at zero degrees eccentricity, in a region known as the fovea (see Fig. 1(a)), their density falling off rapidly with increasing eccentricity, whilst rods are absent from the centre of the fovea, their density increasing sharply with increasing eccentricity, reaching a maximum at around 20° , before falling off more gradually toward the periphery. Thus the fovea is cone dominated, whilst the rest of the retina is rod dominated. Both rods and cones are absent from the optic disc.

Oxygen is delivered to the retina via two blood supplies: the choriocapillaris (CC), a capillary bed forming the innermost layer of a vascular layer known as the choroid, supplies the outer retina, whilst the retinal capillaries mainly supply the inner retina (Wangsa-Wirawan, Linsenmeier, 2003, Yu, Cringle, 2001).

RP usually exhibits as a rod-cone dystrophy, meaning that rods are affected earlier and more severely than cones (Hamel, 2006). However, less common forms exist, in which rod and cone loss occurs on the same timescale, or in which cone loss precedes rod loss (termed a cone-rod dystrophy; Hartong et al., 2006). Histological data from humans and rats suggest that the initial loss of photoreceptors occurs in roughly circular patches which expand and coalesce over time (Cideciyan, Hood, Huang, Banin, Li, Stone, Milam, Jacobson, 1998, García-Ayuso, Ortín-Martínez, Jiménez-López, Galindo-Romero, Cuenca, Pinilla, Vidal-Sanz, Agudo-Barriuso, Villegas-Pérez, 2013, Ji, Zhu, Grzywacz, Lee, 2012, Lee, Ji, Zhu, Grzywacz, 2011, Zhu, Ji, Lee, Grzywacz, 2013). Whilst the initial loss of photoreceptors is due directly to a mutation, it is not known what causes degenerate patches to spread, or what causes cone loss to follow rod loss in the rod-cone dystrophy form (and vice-versa in the cone-rod dystrophy form).

Four main hypotheses have been proposed to explain these phenomena. Firstly, the rod trophic factor hypothesis suggests that rods produce a trophic factor necessary for cone survival, such that, when rods are lost, this factor is depleted, leading to cone loss (Fintz, Audo, Hicks, Mohand-Saïd, Léveillard, Sahel, 2003, Léveillard, Mohand-Saïd, Lorentz, Hicks, Fintz, Clérin, Simonutti, Forster, Cavusoglu, Chalmel, Dollé, Poch, Lambrou, Sahel, 2004, Mohand-Saïd, Deudon-Combe, Hicks, Simonutti, Forster, Fintz, Léveillard, Dreyfus, Sahel, 1998, Mohand-Saïd, Hicks, Dreyfus, Sahel, 2000, Mohand-Saïd, Hicks, Simonutti, Tran-Minh, Deudon-Combe, Dreyfus, Silverman, Ogilvie, Tenkova, Sahel, 1997). Secondly, the toxic substance hypothesis suggests that dying photoreceptors may release toxic substances into their surroundings, poisoning nearby photoreceptors (Ripps, 2002). Thirdly, the microglia hypothesis suggests that microglia cells are activated by rod death to release toxic factors, resulting in the death of the surrounding photoreceptors (Gupta et al., 2003).

In this paper we develop mathematical models to test the fourth hypothesis, known as the oxygen toxicity hypothesis; first suggested by Travis et al. (1991) and later developed by Valter et al. (1998) and Stone et al. (1999). Following an initial loss of photoreceptors, the oxygen demand in the outer retina is significantly decreased, resulting in an increase in outer retinal oxygen concentrations (Padnick-Silver, Derwent, Giuliano, Narfström, Linsenmeier, 2006, Stone, Maslim, Valter-Kocsi, Mervin, Bowers, Chu, Barnett, Provis, Lewis, Fisher, Bistid, Gargini, Cervetto, Merin, Pe’er, 1999, Yu, Cringle, Valter, Walsh, Lee, Stone, 2004, Yu, Cringle, Su, Yu, 2000). These oxygen levels are maintained due to the inability of the choroid to autoregulate in response to increased outer retinal oxygen concentrations (Stone, Maslim, Valter-Kocsi, Mervin, Bowers, Chu, Barnett, Provis, Lewis, Fisher, Bistid, Gargini, Cervetto, Merin, Pe’er, 1999, Yu, Cringle, Valter, Walsh, Lee, Stone, 2004, Yu, Cringle, 2005). This creates a toxic environment for the remaining rod and cone ISs (Stone, Maslim, Valter-Kocsi, Mervin, Bowers, Chu, Barnett, Provis, Lewis, Fisher, Bistid, Gargini, Cervetto, Merin, Pe’er, 1999, Travis, Sutcliffe, Bok, 1991), resulting in a positive feedback loop, as high oxygen levels cause photoreceptor death, which in turn increases oxygen levels. High oxygen levels are toxic to photoreceptors, since they result in an excess of reactive oxygen species, which cause damage to lipids, proteins and DNA (Shen et al., 2005). Studies using human epithelial cells (Wang et al., 2003) and rd1 mouse photoreceptors (Sahaboglu et al., 2013) show that cells take about 72 h to die from hyperoxia. This corresponds to a timescale of decades for degeneration to spread across the retina (given that approximately 4000 photoreceptors lie along any direct path traced through the retina between the ora serrata and the fovea and assuming sequential cell death), a timescale consistent with degeneration in humans (Hartong et al., 2006). Further evidence supporting the role of hyperoxia in RP can be found in Cingolani et al. (2006), Wellard et al. (2005) and Yamada et al. (1999); for reviews, see Roberts (2015), Stone et al. (1999), Yu and Cringle (2001) and Yu and Cringle (2005).

Treatment with antioxidants and trophic factors has been shown to improve photoreceptor survival in RP: antioxidants neutralise the excess of reactive oxygen species produced under hyperoxia (Kohen and Nyska, 2002), whilst trophic factors increase photoreceptor resistance to apoptosis (Dong, Shen, Krause, Akiyama, Hackett, Lai, Campochiaro, 2006, Kohen, Nyska, 2002, Komeima, Rogers, Lu, Campochiaro, 2006, Komeima, Rogers, Campochiaro, 2007, Okoye, Zimmer, Sung, Gehlbach, Deering, Nambu, Hackett, Melia, Esumi, Zack, Campochiaro, 2003, Sanz, Johnson, Ahuja, Ekström, Romero, van Veen, 2007, Yamada, Yamada, Ando, Esumi, Bora, Saikia, Sung, Zack, Campochiaro, 2001, Yu, Cringle, Valter, Walsh, Lee, Stone, 2004).

Degeneration of the CC is commonly observed in human models of RP, although its immediate cause is unknown (Li, Possin, Milam, 1995, Milam, Zong, Fariss, 1998, Mullins, Kuehn, Radu, Enriquez, East, Schindler, Travis, Stone, 2012). The CC is usually absent from areas in which photoreceptors have completely degenerated and in which RPE cells have migrated away from the CC, towards the retinal capillary layers (Li, Possin, Milam, 1995, Milam, Zong, Fariss, 1998), suggesting that the loss of photoreceptors and RPE is either directly or indirectly responsible for CC degeneration (see Roberts, 2015, for a detailed discussion). It has been shown that the CC degenerates within one week after the removal of the RPE (Del Priore et al., 1996); however, to the best of our knowledge, the rate of CC degeneration in RP has not been measured. Since capillary degeneration in the CC leads to a decrease in oxygen supply, this modulates the hyperoxia-driven progression of photoreceptor degeneration. We note that, from this point onwards, we use the term ‘capillaries’ to refer to the capillaries in the CC (rather than the retinal capillaries).

Whilst the precise mechanisms underlying RP have yet to be determined, the spatio-temporal patterns of retinal degeneration are well-defined. In particular, Grover et al. (1998) have identified three distinct patterns of visual field loss in human RP. Pattern 1 involves the concentric loss of vision, starting at the far-periphery, sometimes accompanied by a parafoveal or perifoveal ring scotoma (blind spot), whilst patterns 2 and 3 involve preferential loss of mid-peripheral vision. In all cases the central visual field is preserved, being lost only at the end stage of the disease. Studies by Escher et al. (2012), Lima et al. (2009), Lima et al. (2012), Murakami et al. (2008), Popović et al. (2005), Robson et al. (2003), Robson et al. (2004), Robson et al. (2006), Robson et al. (2008) and Robson et al. (2011) describe similar patterns of degeneration.

To date, few mathematical models have been developed to study retinal degeneration in general and RP in particular (see Roberts et al., 2016, for a detailed review). Colón Vélez et al. (2003), Camacho et al. (2010), Camacho and Wirkus (2013), Camacho et al. (2014), Camacho et al. (2016a), Camacho et al. (2016b) and Camacho et al. (2016c) have produced a series of spatially-averaged ordinary differential equation (ODE) models examining the rod trophic factor hypothesis. Their work suggests the importance of rod trophic factor to the survival of both rods and cones. They were also able to trace the progression of the disease through distinct physiological stages and to suggest optimal treatment strategies for RP under this hypothesis. Building on the experimental work of Clarke et al. (2000) and Clarke et  al. (2001), Burns et al. (2002) constructed a spatially explicit 1D partial differential equation (PDE) model, to examine the toxic substance hypothesis. The model captures the initial patchy loss of photoreceptors as well as the exponential decline in photoreceptor number measured by Clarke, Collins, Leavitt, Andrews, Hayden, Lumsden, McInnes, 2000, Clarke, Lumsden, McInnes, 2001 (see also, Clarke, Lumsden, 2005a, Clarke, Lumsden, 2005b, Lomasko, Clarke, Lumsden, 2007a, Lomasko, Clarke, Lumsden, 2007b, Lomasko, Lumsden, 2009).

In this paper, we use mathematical models to explore, for the first time, the oxygen toxicity hypothesis. Our models combine some of the strengths of previous modelling work, in that they are spatially explicit, and distinguish between rods and cones. In addition, they are the first to account for the heterogeneous distribution of rods and cones, and its effects upon the spatio-temporal pattern of degeneration.

We use our models to determine if photoreceptor degeneration could spread due to hyperoxia; the conditions under which a degenerate patch of photoreceptors will remain stable and under which it will spread; which spatio-temporal patterns of progression are possible or most likely via this mode of degeneration; the variation in the wave speed of degeneration with eccentricity; and the effects of treatment with antioxidants and trophic factors, and of capillary loss upon photoreceptor degeneration.

The remainder of this paper is structured as follows: in Section 2 we develop the models, in Section 3 we suppose that the capillaries remain healthy and examine degenerate patch stability (Sections 3.1 and 3.2), the wave speed of degeneration (Sections 3.3 and 3.4), the effects of mutation-induced rod and/or cone loss (Sections 3.5 and 3.6), and the effects of treatment with antioxidants and trophic factors (Section 3.7). In Section 4 we consider the effects of capillary degeneration upon photoreceptor loss. Lastly, in Section 5 we discuss our results and suggest directions for future research.

Section snippets

Model formulation

We view the posterior of the eye as a spherical cap and describe the geometry of the retina using spherical polar coordinates, (r, θ, ϕ), where θ is the polar angle and ϕ is the azimuthal angle (see Fig. 1). Since the part of the eye containing the retina is almost spherical (Oyster, 1999) this is a reasonable approximation. The fovea lies almost exactly at the centre of the retina, opposite the lens, and so, ignoring the optic disc, we assume that the retina is axisymmetric about the z-axis

Results: Model 1 — no capillary degeneration

In this section, we consider the dynamics of photoreceptor loss when capillary degeneration is neglected, i.e. the capillary surface area per unit volume of tissue is given by h(θ, t) ≡ 1, rather than the dynamics summarised by Eqs. (5), (10) and (16) for the respective models.

Results: Model 2 — capillary degeneration

Thus far we have assumed that the capillary surface area per unit volume of tissue, h(θ, t), remains constant over time. In this section, we examine the effects of capillary loss upon the dynamics of photoreceptor degeneration.

Discussion

RP causes the progressive degeneration of photoreceptors. In humans, photoreceptor loss initiates in patches, which spread and coalesce until only a central island of photoreceptors remain, these also eventually being lost. RP typically exhibits as a rod-cone dystrophy, in which rod photoreceptors are affected earlier and more severely than cone photoreceptors; however, cone-rod dystrophies also occur, in which the reverse is true. It is not known what causes degenerate patches to spread, or

Acknowledgements

We gratefully acknowledge the Engineering and Physical Sciences Research Council (EPSRC) in the UK for funding through a studentship at the Systems Biology programme of the University of Oxford’s Doctoral Training Centre P.A.R. We also thank the anonymous reviewers for their helpful and insightful comments.

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    Present address: School of Mathematics, University of Birmingham, Edgbaston Campus, Birmingham, B15 2TT, UK

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